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Chapter 3: Problem 8

Determine whether each equation is linear. $$ 2 y+10 x=5 $$

Short Answer

Expert verified
Yes, the equation is linear as the variables are all of degree 1.

Step by step solution

01

- Identify Variables and Constants

Identify the variables (e.g., x and y) and constant terms in the equation. The given equation is: off by navigating to simpler forms like \( y = x + c_{onstant}\) and \( m or adjusting `\displaystyle \text{Right} \ Right` language inference. \)2y + 10x = 5 $. Here, `y and x` are the variables, whereas `5` is the constant term.
02

- Check the Degree of the Variables

In a linear equation, each term should be either a constant or a product of a constant and a single variable raised to the first power (degree of 1). In the given equation: The term 10x and 2y are containing` single variables of`first power all the terms contain the variables with a degree of 1.
03

Conclusion

Since all variables in the equation have a degree of 1 this indicates that the given equation is linear.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variables
Variables are symbols used to represent unknown values. In the equation provided, \(2y + 10x = 5\), the variables are \(x\) and \(y\).
Variables can change their value depending on the conditions of the equation. They are essential in forming equations and are often accompanied by constants.
Understanding variables is crucial because they are the building blocks of algebraic expressions and equations. Without variables, we cannot form equations or solve for unknown values.
Constants
Constants are fixed values that do not change. In the equation \(2y + 10x = 5\), the number \(5\) is a constant.
Unlike variables, constants remain the same regardless of the values of \(x\) or \(y\).
Recognizing constants is important because they help balance equations and provide a reference point when solving for the variables. They also aid in determining the form of the equation, whether it is linear or not.
Degree of Variables
The degree of a variable in a term is the exponent to which the variable is raised. In a linear equation, each variable must have a degree of \(1\).
For example, in the term \(10x\), \(x\) has an implied exponent of \(1\). Similarly, in the term \(2y\), \(y\) also has an exponent of \(1\).
If any variable in the equation had an exponent greater than \(1\) (e.g., \(x^2\) or \(y^3\)), the equation would no longer be linear.
Identifying the degree of variables helps confirm whether an equation is linear or not.
Linear Equation Identification
A linear equation is an equation where all terms are either constants or products of constants and variables with a degree of \(1\).
To identify a linear equation, follow these steps:
  • Locate the variables and constants in the equation.
  • Check that each variable has a degree of \(1\).
  • Ensure that there are no variables with exponents greater than \(1\).

For the given equation \(2y + 10x = 5\), we have variables \(x\) and \(y\), both with a degree of \(1\), and a constant \(5\). Therefore, we identify this as a linear equation.

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Most popular questions from this chapter

A Boeing 737 climbs from sea level to a cruising altitude of \(31,500\) ft at a rate of \(6300 \mathrm{ft} / \mathrm{min}\). After cruising for 3 min, the jet is forced to land, descending at a rate of \(3500 \mathrm{ft} / \mathrm{min} .\) Represent the flight with a graph in which altitude is measured on the vertical axis and time on the horizontal axis.

Find the function values. $$s(x)=\frac{3 x-4}{2 x+5}$$ a) \(s(10)\) b) \(s(2)\) c) \(s\left(\frac{1}{2}\right)\) d) \(s(-1)\) e) \(s(x+3)\)

Write a slope-intercept equation of the line whose graph is described. Parallel to the graph of \(3 x+2 y=5 ; y\) -intercept \((0,-1)\)

Simplify. $$-2^{6}$$

To prepare for Section \(3.5,\) review subtraction and order of operations. Simplify. $$ \frac{-3-5}{-1-(-1)} $$
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